Unit 4B Day 3 and 4

Notes on CPCTC

and proofs with overlapping triangles

Day 3 Warm-Up

Then do # 26, 28, 30 from yesterday’s Proof Packet

Day 3 Warm-Up Answers

HW Discussion

Proving Parts of Triangles Congruent:

With SSS, SAS, ASA, and AAS, you know how to use three parts

of triangles to show that triangles are congruent.

Once you have triangles congruent, you can make conclusions

about their other parts.

By definition, corresponding parts of congruent triangles are congruent.

You can abbreviate this as CPCTC

What other pairs of sides and angles can you conclude are

congruent by CPCTC?

Ex 1.

Given:

Prove:

R Q

QPS RSP

PR SQ

P Q

R S

Statement Reason

1)∠𝑅 ≅ ∠𝑄

∠𝑄𝑃𝑆 ≅ ∠𝑅𝑆𝑃1) Given

2) 𝑃𝑆 ≅ 𝑃𝑆 2) Reflexive property of

congruence

3) ∆𝑄𝑃𝑆 ≅ ∆𝑅𝑆𝑃 3) AAS Theorem

4) 𝑃𝑅 ≅ 𝑆𝑄 4) CPCTC

According to legend,one of Napoleon’s officer usedcongruent triangles to estimatethe width of a river. On theriverbank, the officer stood upstraight and lowered the visorof his cap until the farthest thinghe could see was the edge of theopposite bank. He then turnedand noted the spot on his sideof the river that was in line withhis eye and the tip of his visor.The officer then paced off the distance to this spot and declared that distance to bethe width of the river! Use congruent triangles to prove that he was correct.

Given: DEG and DEF are right angles

EDG EDF

Prove:

EG EF (proof on next slide)

Given: DEG and DEF are right angles

EDG EDF

Prove:

EG EF

Statement Reasons

1. EDG EDF 1. Given

2. DE DE 2. Reflexive Property of Congruence

3. DEG and DEF are 3. Given

right angles.

4. DEG DEF 4. All right angles are congruent.

5. ∆DEF ∆ DEG 5. ASA Postulate

6. EG EF 6. CPCTC

What other congruence statements can you prove from the

diagram, in which SL SR, and 1 2 are given?

SC SC by the Reflexive Property of Congruence,

and ΔLSC Δ RSC by SAS Postulate.

Then, 3 4 because corresponding parts of congruent

triangles are congruent.

When two triangles are congruent,

you can form congruence

statements about three pairs of

corresponding angles and three

pairs of corresponding sides.

You could also use CPCTC to prove

CLS CRS and CL CR

Practice

1) Notes online p. 16 #11, 12, 15

2) Bottom 4 Congruent Triangle

Challengers from

http://feromax.com/cgi-bin/ProveIt.pl

3) Notes online p. 16-18 #20, 21

Some triangle relationships are difficult to see because the triangles overlap. Overlapping triangles may have a common side or angle.

You can simplify your work with overlapping triangles by separating and redrawing the triangles.

Ex: Name the parts of their sides that ΔDFG and ΔEHG share.

1st, Identify the overlapping triangles.

2nd, Identify the shared parts.

These are HG and FG, respectively,and angle G.

More tips on the next slide!

Draw the triangles separately. Using colors can help! Trace the triangles on the original diagram. Look at where the colors overlap.

Then, look at the amount of repeated letters between the two triangles.• Often, 1 repeated letter means there is an angle pair congruent by reflexive property.• Often, 2 repeated letters means there is a side pair congruent by reflexive property.

Practice

Continue work on online

Notes p. 15-18

Exit Ticket:1.What does “CPCTC” stand for?

Use the diagram for Exercises 2 and 3.

2. Tell how you would show

ABM ≅ ACM.

3. Tell what other parts are congruent

by CPCTC.

Use the diagram for Exercise 4.

4. Given: <Q ≅ <S, and RU ≅ TU

Prove: RQ ≅ TS.

1. What does “CPCTC” stand for?

Use the diagram for Exercises 2 and 3.2. Tell how you would showABM ≅ ACM.

3. Tell what other parts are congruent by CPCTC.

Use the diagram for Exercises 4 and 5.

4. Given: <Q ≅ <S, and RU ≅ TU

Prove: RQ ≅ TS.

You are given two pairs of ≅s and AM ≅ AM by the Reflexive Prop., so ABM ≅ ACM by ASA.

Corresponding parts of congruent triangles are congruent.

AB ≅ AC, BM ≅ CM, B ≅ C

1) <Q ≅ <S, 1) Given

and RU ≅ TU

2) RUQ ≅TUS 2) vertical angles are ≅3) RUQ ≅ TUS 3) AAS Theorem

4) 4) CPCTCRQ ≅ TS .

Day 3 Collected Proof:Requirements: Write down ALL parts of the problem and mark in your

picture. No Talking! This proof will be graded for accuracy!

Given: AB≌CD

AB║CD

Prove: ABC ≅ CDA

A B

D C

Statements Reasons

Day 3 Collected Proof:Requirements: Write down ALL parts of the problem and mark in your

picture. No Talking! This proof will be graded for accuracy!

Given: AB≌CD

AB║CD

Prove: ABC ≅ CDA

A B

D C

Statements Reasons

Day 4

Practice with Proofs

Day 4 Warm-Up

Prove that

Day 4 Warm-Up Answers

Day 4 Collected Proof Answers:Requirements: Write down ALL parts of the problem and mark in your

picture. No Talking! This proof will be graded for accuracy!

Given: AB≌CD

AB║CD

Prove: 𝐴𝐷 ≅ 𝐶𝐵

A B

D C

Statements Reasons

Day 4 Collected Proof:Requirements: Write down ALL parts of the problem and mark in your

picture. No Talking! This proof will be graded for accuracy!

Given: AB≌CD

AB║CD

Prove: 𝐴𝐷 ≅ 𝐶𝐵

A B

D C

Statements Reasons

HW Discussion

A tougher CPCTC problem

Complete a 2-column proof for….

2) Given: , PSQ RSQProve:

PS RS

QPT QRTP

Q

R

S

T

2) Given: , PSQ RSQProve:

PS RS

QPT QRT

P

Q

R

S

T

Statement Reason

𝑃𝑆 ≅ 𝑅𝑆, PSQ RSQ Given

𝑆𝑄 ≅ 𝑆𝑄 Reflexive property of congr.

ΔPSQ ΔRSQ SAS Post.

𝑃𝑄 𝑅𝑄 and

PQS RQS

CPCTC

𝑄𝑇 ≅ 𝑄𝑇 Reflexive property of congr.

𝛥𝑄𝑃𝑇 ≅ 𝛥𝑄𝑅𝑇 SAS Post.

Answer to tougher CPCTC problem

E

A

B C

D

Practice Proof – tougher one using CPCTC

E

A

B C

D

Practice Proofs – tougher one using CPCTC

Statement Reason

𝐴𝐵 𝐵𝐶, 𝐷𝐶 𝐵𝐶, 𝐴𝐶 ≅ 𝐷𝐵 Given

ABC and DCB are Right Angles Defn. of perpendicular

ABC and DCB are Right ∆s Defn. of Right Triangle

𝐵𝐶 ≅ 𝐵𝐶 Reflexive Property of ≅∆𝐴𝐵𝐶 ≅ ∆𝐷𝐶𝐵 HL Theorem

𝐴𝐵 ≅ 𝐷𝐶 𝑎𝑛𝑑 𝐴 ≅ 𝐷 CPCTC

𝐴𝐸𝐵 ≅ 𝐷𝐸𝐶 Vertical angles Theorem

∆𝐴𝐸𝐵 ≅ ∆𝐷𝐸𝐶 AAS Theorem

𝐴𝐸 ≅ 𝐷𝐸 CPCTC

Practice

Continue work on online

Notes p. 15-18

Complete a 2-column proof.

Given:

Prove:

ADis a bisector of BC

AB AC

Day 4 Collected Proof:

Requirements: Write down ALL

parts of the problem and mark

in your picture. No Talking!

B

D

C

A

Given:

Prove:

ADis a bisector of BC

AB AC

Day 4 Collected Proof Answer:B

D

C

A

Statement Reason

Given

ADC and ADB are right s Defn. of perpendicular lines

ADC≅ ADB All Right angles are congruent

≅ Defn. of bisector

≅ Reflexive Property of

congruence

ADC≅ ADB SAS Postulate

CPCTC

BD CD

AD AD

AB AC